I tried to make a V2 of TJK 31 (Ed askes for one ). Because the cage pattern combined wih the jigsaw shapes gave away quick singles i tried to change a few cages to make the opening less obvious.
These both use the same solution as the original.

TJK 31V1.5
This one seems to be a bit harder. But there was a different opening i missed in the first run. Which in the end doesn't make it much harder than the original.

TJK 31V2
This one is a lot harder. That is why i included the V1.5. It is more fun to solve.

I don't know how to get a PS-string for jigsaw killers from SumoCue. Maybe someone else can provide these.

A few eliminations! Hope these are correct.
31. no 8 in r8c78, Here's how. (This might be easier to see if they were written as xy chains: might have to edit)
31a. r7c456 = same combinations as 15(3)r6c2 (from LoL r789)
31b. = [1]{59}/[2]{58}/[3]{57}
31c. -> r7c4 + r7c56 + r7c7 = [1]{59}[6]/[2]{58}[7]/[3]{57}[8]
31d. -> r7c7 + r8c4 = [68/77/86]
i. 8 is in r8c4 when 6 is in r7c7 -> no 8 in r8c78
ii. or 8 is in r9c7 when r7c7 = 7 -> no 8 in r8c78
iii. or 8 is in r7c7 -> no 8 in r8c78

32. no 7 in r8c12. Here's how.
32a. from LoL r89: 3 outies r7c789 = 3 innies r8c12 + r9c1
32b. 7 in outies in r7c7 -> 7 in r8c4 (step 31d) -> no 7 in r8c12
-> 7 in innies only fits in r9c1.
32c. if 7 is not in r7c7 -> from LoL r89, no 7 is possible in 8c12